### Nuprl Lemma : length_concat

`∀[T:Type]. ∀[ll:T List List].  (||concat(ll)|| = Σ(||ll[i]|| | i < ||ll||) ∈ ℤ)`

Proof

Definitions occuring in Statement :  sum: `Σ(f[x] | x < k)` select: `L[n]` length: `||as||` concat: `concat(ll)` list: `T List` uall: `∀[x:A]. B[x]` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` int_seg: `{i..j-}` uimplies: `b supposing a` guard: `{T}` lelt: `i ≤ j < k` and: `P ∧ Q` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` prop: `ℙ` less_than: `a < b` squash: `↓T` so_apply: `x[s]` concat: `concat(ll)` select: `L[n]` nil: `[]` it: `⋅` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` sum: `Σ(f[x] | x < k)` sum_aux: `sum_aux(k;v;i;x.f[x])` subtype_rel: `A ⊆r B` nat: `ℕ` le: `A ≤ B` less_than': `less_than'(a;b)` ge: `i ≥ j ` uiff: `uiff(P;Q)` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` sq_type: `SQType(T)`
Lemmas referenced :  list_induction list_wf equal_wf length_wf concat_wf sum_wf length_wf_nat select_wf int_seg_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma int_seg_wf reduce_nil_lemma length_of_nil_lemma stuck-spread base_wf length_of_cons_lemma concat-cons squash_wf true_wf length_append subtype_rel_list top_wf add_nat_wf false_wf le_wf nat_wf nat_properties add-is-int-iff itermAdd_wf intformeq_wf int_term_value_add_lemma int_formula_prop_eq_lemma cons_wf non_neg_length iff_weakening_equal subtype_base_sq int_subtype_base sum_split lelt_wf sum1 select-cons-hd decidable__equal_int subtract_wf itermSubtract_wf int_term_value_subtract_lemma select-cons-tl add-subtract-cancel
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin extract_by_obid sqequalHypSubstitution isectElimination cumulativity hypothesisEquality hypothesis sqequalRule lambdaEquality intEquality because_Cache setElimination rename independent_isectElimination natural_numberEquality productElimination dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll imageElimination independent_functionElimination baseClosed lambdaFormation applyEquality equalityTransitivity equalitySymmetry equalityUniverse levelHypothesis dependent_set_memberEquality addEquality applyLambdaEquality pointwiseFunctionality promote_hyp baseApply closedConclusion imageMemberEquality universeEquality instantiate functionEquality axiomEquality

Latex:
\mforall{}[T:Type].  \mforall{}[ll:T  List  List].    (||concat(ll)||  =  \mSigma{}(||ll[i]||  |  i  <  ||ll||))

Date html generated: 2017_04_17-AM-08_50_25
Last ObjectModification: 2017_02_27-PM-05_07_32

Theory : list_1

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